Random Matrix Models and their Applications

Mathematical Sciences Research Institute

doi:10.1017/9781009701440

Random matrices arise from, and have important applications to, number theory, probability, combinatorics, representation theory, quantum mechanics, solid state physics, quantum field theory, quantum gravity, and many other areas of physics and mathematics. This 2001 volume of surveys and research results, based largely on lectures given at the Spring 1999 MSRI program of the same name, covers broad areas such as topologic and combinatorial aspects of random matrix theory; scaling limits, universalities and phase transitions in matrix models; universalities for random polynomials; and applications to integrable systems. Its stress on the interaction between physics and mathematics will make it a welcome addition to the shelves of graduate students and researchers in both fields, as will its expository emphasis.

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Contents

Frontmatter
pp i-vi
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Contents
pp vii-viii
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Preface
pp ix-x
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Symmetrized Random Permutations
pp 1-20
- By Jinho Baik, Eric M. Rains
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Hankel Determinants as Fredholm Determinants
pp 21-30
- By Estelle L. Basor, Yang Chen, Harold Widom
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Universality and Scaling of Zeros on Symplectic Manifolds
pp 31-70
- By Pavel Bleher, Bernard Shiffman, Steve Zelditch
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z-Measures on Partitions, Robinson-Schensted-Knuth Correspondence, and β = 2 Random Matrix Ensembles
pp 71-94
- By Alexei Borodin, Grigori Olshanski
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Phase Transitions and Random Matrices
pp 95-110
- By Giovanni M. Cicuta
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Matrix Model Combinatorics: Applications to Folding and Coloring
pp 111-170
- By Philippe Di Francesco
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Interrelationships Between Orthogonal, Unitary and Symplectic Matrix Ensembles
pp 171-208
- By Peter J. Forrester, Eric M. Rains
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Dual Isomonodromic Tau Functions and Determinants of Integrable Fredholm Operators
pp 209-224
- By J. Harnad
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Functional Equations and Electrostatic Models for Orthogonal Polynomials
pp 225-244
- By Mourad E. H. Ismail
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Random Words, Toeplitz Determinants, and Integrable Systems I
pp 245-258
- By Alexander R. Its, Craig A. Tracy, Harold Widom
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Random Permutations and the Discrete Bessel Kernel
pp 259-270
- By Kurt Johansson
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Solvable Matrix Models
pp 271-284
- By Vladimir Kazakov
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The τ-Function for Analytic Curves
pp 285-300
- By I. K. Rostov, I. Krichever, M. Mineev-Weinstein, P. B. Wiegmann, A. Zabrodin
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Integration over Angular Variables for Two Coupled Matrices
pp 301-320
- By G. Mahoux, M. L. Mehta, J.-M. Norm
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Integrable Lattices: Random Matrices and Random Permutations
pp 321-406
- By Pierre Van Moerbeke
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SL(2) and z-Measures
pp 407-420
- By Andrei Okounkov
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Some Matrix Integrals Related to Knots and Links
pp 421-438
- By Paul Zinn-Justin
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Random Matrix Models and their Applications

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Contents

Frontmatter
pp i-vi

Contents
pp vii-viii

Preface
pp ix-x

Symmetrized Random Permutations
pp 1-20

Hankel Determinants as Fredholm Determinants
pp 21-30

Universality and Scaling of Zeros on Symplectic Manifolds
pp 31-70

z-Measures on Partitions, Robinson-Schensted-Knuth Correspondence, and β = 2 Random Matrix Ensembles
pp 71-94

Phase Transitions and Random Matrices
pp 95-110

Matrix Model Combinatorics: Applications to Folding and Coloring
pp 111-170

Interrelationships Between Orthogonal, Unitary and Symplectic Matrix Ensembles
pp 171-208

Dual Isomonodromic Tau Functions and Determinants of Integrable Fredholm Operators
pp 209-224

Functional Equations and Electrostatic Models for Orthogonal Polynomials
pp 225-244

Random Words, Toeplitz Determinants, and Integrable Systems I
pp 245-258

Random Permutations and the Discrete Bessel Kernel
pp 259-270

Solvable Matrix Models
pp 271-284

The τ-Function for Analytic Curves
pp 285-300

Integration over Angular Variables for Two Coupled Matrices
pp 301-320

Integrable Lattices: Random Matrices and Random Permutations
pp 321-406

SL(2) and z-Measures
pp 407-420

Some Matrix Integrals Related to Knots and Links
pp 421-438

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Random Matrix Models and their Applications

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